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Keywords:
exponential sum; multiplicative arithmetic function; hereditary sum of digits function
exponential sum; multiplicative arithmetic function; hereditary sum of digits function
Summary:
The aim of this paper is to conduct a survey on the average of some selected multiplicative arithmetic functions for large values of integers, while considering constraints imposed by the hereditary sum of digits function in base $b$, denoted by $w_{b}$. This investigation aims to provide insights into the behavior of these functions within the scope of the hereditary sum of digits.
References:
[1] Aloui K.: On the order of magnitude of some arithmetical functions under digital constraint I. Proc. Indian Acad. Sci. Math. Sci. 125 (2015), no. 4, 457–476. DOI 10.1007/s12044-015-0231-x
[2] Aloui K.: On the average of some arithmetical functions under a constraint on the sum of digits of squares. Hiroshima Math. J. 46 (2016), no. 1, 37–54. DOI 10.32917/hmj/1459525929
[3] Aloui K.: On the order of magnitude of some arithmetical functions under digital constraint II. Math. Rep. (Bucur.) 24 (74) (2022), no. 3, 403–423.
[4] Aloui K., Feki F.: On the distribution of the integers with missing digits under hereditary sum of digits. Publ. Math. Debrecen 94 (2019), no. 3–4, 337–358. DOI 10.5486/PMD.2019.8295
[5] Amri M., Mbarki K.: Mean values of arithmetic functions under congruences with the Euler function. Turkish Journal of Analysis and Number Theory 8 (2020), no. 2, 39–48. DOI 10.12691/tjant-8-2-4
[6] Andrica D., Piticari M.: On some extensions of Jordan's arithmetic functions. Acta Univ. Apulensis Math. Inform. 7 (2004), 13–22.
[7] Apostol T. M.: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, Springer, New York, 1976.
[8] Balakrishnan U., Pétermann Y.-F. S.: Asymptotic estimates for a class of summatory functions. II. Comment. Math. Univ. St. Pauli 50 (2001), no. 1, 89–109.
[9] Bésineau J.: Indépendance statistique d'ensembles liés à la fonction “somme des chiffres". Acta Arith. 20 (1972), 401–416 (French). DOI 10.4064/aa-20-4-401-416
[10] Coquet J.: Sur les fonctions $Q$-multiplicatives et $Q$-additives. Thèse 3 cycle, Orsay, 1975 (French).
[11] Delange H.: Sur les fonctions $q$-additives ou $q$-multiplicatives. Acta Arith. 21 (1972), no. 1, 285–298. DOI 10.4064/aa-21-1-285-298
[12] Drmota M., Mauduit C., Rivat J.: Primes with an average sum of digits. Compos. Math. 145 (2009), no. 2, 271–292. DOI 10.1112/S0010437X08003898
[13] Feki F.: On the hereditary sum of digits function to base $q$. J. Ramanujan Math. Soc. 36 (2021), no. 2, 157–168.
[14] Goodstein R. L.: On the restricted ordinal theorem. J. Symbolic Logic 9 (1944), no. 2, 33–41. DOI 10.2307/2268019
[15] Halberstam H., Richert H.-E.: Mean value theorems for a class of arithmetic functions. Acta Arith. 18 (1971), no. 1, 243–256. DOI 10.4064/aa-18-1-243-256
[16] Hardy G. H., Wright E. M.: An Introduction to the Theory of Numbers. The Clarendon Press, Oxford University Press, New York, 1979.
[17] Ikehara S.: An extension of Landau's theorem in the analytical theory of numbers. J. of Math. Phys. 10 (1931), no. 1–4, 1–12. DOI 10.1002/sapm19311011
[18] Jemai A.: On the average of some multiplicative functions under a constraint on joint Zeckendorf representation. Hiroshima Math. J. 55 (2025), no. 1, 1–21. DOI 10.32917/h2023005
[19] Kátai I.: Distribution of $q$-additive functions. in Probability Theory and Applications, Math. Appl., 80, Kluwer Acad. Publ., Dordrecht, 1992, pages 309–318.
[20] Kirby L., Paris J.: Accessible independence results for Peano arithmetic. Bull. Lond. Math. Soc. 14 (1982), no. 4, 285–293. DOI 10.1112/blms/14.4.285
[21] Lehmer D. H.: Euler constants for arithmetical progressions. Acta Arith. 27 (1975), 125–142. DOI 10.4064/aa-27-1-125-142
[22] Levin B. V., Faĭnleĭb A. S.: Application of certain integral equations to questions of the theory of numbers. Uspehi Mat. Nauk 22 (1967), no. 3(135), 119–197 (Russian).
[23] van Lint J. H., Richert H.-E.: On primes in arithmetic progressions. Acta Arith. 11 (1965), 209–216. DOI 10.4064/aa-11-2-209-216
[24] Mbarki K., Wannes W.: Means of some arithmetical functions on shifted smooth numbers in arithmetic progression. Proc. Indian Acad. Sci. Math. Sci. 129 (2019), no. 4, Paper No. 53, 22 pages. DOI 10.1007/s12044-019-0521-9
[25] Sanna C.: On the exponential sum with the sum of digits of hereditary base $b$ notation. Integers 14 (2014), Paper No. A36, 10 pages.
[26] Tóth L.: A Survey of gcd-sum functions. J. Integer Seq. 13 (2010), no. 8, Article 10.8.1, 23 pages.
[27] Wiener N.: A note on Tauberian theorems. Ann. of Math. (2) 33 (1932), no. 4, 787. DOI 10.2307/1968223
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